Tories will find that the "game is up" at the next general election if they are divided and care more about who leads their party than who runs Britain, the former Conservative chairman Lord Patten of Barnes has warned.

As an ally of George Osborne became the first minister to confirm in public that Theresa May is laying the ground for a leadership bid, Patten – who is also chairman of the BBC – called on his party not to lose sight of the importance of unity.

The remarks by Patten, at a lunch for journalists at Westminster, came as the business minister Michael Fallon confirmed that Downing Street believes that the home secretary is making elaborate plans to stand for the Tory leadership.

Fallon, who is a regular visitors to Nos 10 and 11 Downing Street, joked about May's ambitions in the commons when the veteran Labour MP John Spellar said that the home secretary "might even help her leadership ambitions" if she persuaded police forces to buy more British vehicles. Fallon replied: "They may not need that much help."

Patten said: "I think parties which aren't totally united don't win elections. And what used to be one of the most important characteristics of the Conservative party, and I hope still is, it was more interested in running the country than who ran the Conservative Party. I think if you lose sense of being united, if people start to think you're more interested in fighting one another than fighting for the country, then the game is up."

May said she feels the party needs to do more to reassure people that it poses no threat to public services: "We have to reassure people about our motives and our values. We have to show them that we're committed to universal public services and we must never appear to be doctrinaire about what we're doing. We have to be clear that we're reducing spending on public services not because we care little for them, but because we have to for the sake of our economy."